Quotient of complex numbers in polar form calculator
Dividing 8∠150° by 2∠30° gives 4∠120°, or −2 + 3.464101615138i. Enter each complex number as a modulus and an angle, in degrees or radians, and get the quotient in polar form and as real and imaginary parts. Free in your browser.
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Worked example
- Numerator modulus
- 8
- Numerator argument
- 150
- Denominator modulus
- 2
- Denominator argument
- 30
- Angle unit
- Deg
- Quotient modulus
- 4
- Argument difference
- 120
- Quotient argument
- 120
- Polar form
- 4∠120°
This quotient of complex numbers in polar form calculator performs the operation that polar notation makes especially simple: it divides the numerator modulus by the denominator modulus and subtracts the denominator argument from the numerator argument. Enter both complex numbers as a modulus and an argument, choose degrees or radians, and receive the quotient as a normalized polar value plus its real and imaginary components. The calculation is deterministic, runs without network access, and keeps the intermediate argument difference visible so you can check every part of the result.
Why division is simpler in polar form
A nonzero complex number can be written in polar form as r∠θ, or equivalently as r(cos θ + i sin θ), where r is its nonnegative modulus and θ is an argument. When one polar complex number is divided by another, the quotient rule separates neatly into two ordinary operations: divide the moduli and subtract the arguments. For a numerator r₁∠θ₁ and denominator r₂∠θ₂, the result is (r₁/r₂)∠(θ₁−θ₂), provided r₂ is greater than zero.
This works because division reverses multiplication: magnitudes scale by the reciprocal modulus, while rotations reverse by the denominator angle. The calculator follows that identity directly and does not first convert both inputs to rectangular form. It reports the raw argument difference so the subtraction remains auditable, then also reports an equivalent principal argument.
Since angles differing by a whole turn identify the same complex number, normalization changes the displayed angle but never changes the value. A zero numerator is allowed and produces zero, although its displayed argument is only a conventional representative; a zero denominator is rejected because division by zero is undefined.
Checking signs, angle units, and equivalent answers
Most mistakes in polar division come from reversing the angle subtraction or mixing degrees with radians. Keep the numerator first in both operations: divide numerator modulus by denominator modulus, and calculate numerator argument minus denominator argument. For example, (8∠150°)/(2∠30°) has modulus 4 and argument 120°, so its rectangular value is −2 + 2√3i, approximately −2 + 3.464101615138i.
If you accidentally subtract in the other direction, you get −120°, which is generally the conjugate rather than the quotient. The algorithm uses only finite arithmetic and trigonometric functions, with no random choices, external services, stored data, or dependence on the current time.
What you can do with it
Check complex-number homework
Verify modulus division, argument subtraction, principal-angle normalization, and the final Cartesian coordinates in one response.
Divide engineering phasors
Compute ratios of voltage, current, impedance, or frequency-domain values while retaining both magnitude-phase and real-imaginary forms.
FAQ
What is the rule for dividing complex numbers in polar form?
Divide the numerator modulus by the denominator modulus and subtract the denominator argument from the numerator argument: (r₁∠θ₁)/(r₂∠θ₂) = (r₁/r₂)∠(θ₁−θ₂).
How do I divide 8∠150° by 2∠30°?
Divide the moduli, 8 by 2, to get 4, and subtract the angles, 150° minus 30°, to get 120°. The quotient is 4∠120°, which in rectangular form is −2 + 3.464101615138i.
What happens if I subtract the angles in the wrong order?
You get −120° instead of 120°, which generally gives the conjugate rather than the quotient. Always take the numerator's angle minus the denominator's, and keep the numerator first when dividing the moduli too.
Can I divide by a complex number with a modulus of zero?
No. Division by zero is undefined, so a zero denominator is refused. A zero numerator is allowed and simply gives zero, although its displayed angle is only a conventional placeholder.
Why does my textbook show a different angle for the same result?
Angles that differ by a whole turn, 360° or 2π, describe the same complex number. The calculator shows the direct difference and an equivalent principal angle, so both can be right.
Related tools
For developers — API access
Everything on this page is available programmatically. This section is for teams who want to wire it into their own systems; everyone else can just use the tool above.
API endpoint
Prefer to automate it? One authenticated POST creates the task; the result comes back by webhook or a signed link. The same capability also runs here on the web, by email and from Telegram — and soon from our app too.
Call it from your stack
curl -X POST https://api.kit.forhosting.com/trig/complex-quotient-polar \
-H "Authorization: Bearer $KIT_KEY" \
-H "Content-Type: application/json" \
-d '{"numerator_modulus":8,"numerator_argument":150,"denominator_modulus":2,"denominator_argument":30}'const res = await fetch("https://api.kit.forhosting.com/trig/complex-quotient-polar", {
method: "POST",
headers: {
"Authorization": `Bearer ${process.env.KIT_KEY}`,
"Content-Type": "application/json"
},
body: JSON.stringify({
"numerator_modulus": 8,
"numerator_argument": 150,
"denominator_modulus": 2,
"denominator_argument": 30
})
});
const { task_id } = await res.json();import os, requests
res = requests.post(
"https://api.kit.forhosting.com/trig/complex-quotient-polar",
headers={"Authorization": f"Bearer {os.environ['KIT_KEY']}"},
json={
"numerator_modulus": 8,
"numerator_argument": 150,
"denominator_modulus": 2,
"denominator_argument": 30
},
)
task_id = res.json()["task_id"]<?php
$res = file_get_contents("https://api.kit.forhosting.com/trig/complex-quotient-polar", false, stream_context_create([
"http" => [
"method" => "POST",
"header" => "Authorization: Bearer " . getenv("KIT_KEY") . "\r\nContent-Type: application/json",
"content" => '{"numerator_modulus":8,"numerator_argument":150,"denominator_modulus":2,"denominator_argument":30}',
],
]));
$task = json_decode($res, true);body := bytes.NewBufferString(`{"numerator_modulus":8,"numerator_argument":150,"denominator_modulus":2,"denominator_argument":30}`)
req, _ := http.NewRequest("POST", "https://api.kit.forhosting.com/trig/complex-quotient-polar", body)
req.Header.Set("Authorization", "Bearer "+os.Getenv("KIT_KEY"))
req.Header.Set("Content-Type", "application/json")
res, _ := http.DefaultClient.Do(req)Example request
{
"numerator_modulus": 8,
"numerator_argument": 150,
"denominator_modulus": 2,
"denominator_argument": 30
}Example response
{
"task_id": "tsk_a1b2c3d4e5f6a1b2c3d4e5f6",
"type": "trig.complex_quotient_polar",
"status": "queued",
"_links": {
"result": "/tasks/tsk_…/result"
}
}The API is asynchronous: the call returns a task_id immediately and the result arrives by webhook. Polling is capped at 1 req/s per task.
Pricing
Published price — no tokens, no invented credits. A failed task is never charged.
Errors
| HTTP | Code | Meaning |
|---|---|---|
401 | unauthorized | Missing or invalid API key. |
402 | insufficient_balance | Your balance doesn't cover the task price. |
404 | unknown_type | That task type doesn't exist. |
429 | rate_limited | Too many requests. Use the webhook instead of polling. |
Technical notes
How to enter and interpret the quotient
Floating-point values are rounded to twelve decimal places for stable responses, so tiny trigonometric artifacts such as 2.4 × 10⁻¹⁶ near an axis normally become zero. Enter numerator_modulus and denominator_modulus as finite numbers using the usual polar convention: moduli cannot be negative, and the denominator modulus must be strictly positive. Enter numerator_argument and denominator_argument in the same unit, then select deg for degrees or rad for radians. The output field argument_difference is the direct subtraction θ₁−θ₂ in that selected unit. The quotient_argument field is the equivalent principal angle, normalized to a standard interval around zero, which makes results easier to compare even when the input angles contain several complete revolutions. The quotient_modulus field is r₁/r₂. The polar_form string combines that modulus and normalized argument for convenient reading, while real and imaginary give the Cartesian coordinates computed as r cos θ and r sin θ. The rectangular_form string presents those coordinates as a + bi with the sign formatted explicitly. A successful API calculation costs $0.002; validation failures identify malformed or mathematically undefined inputs.
Checking signs, angle units, and equivalent answers
Also make certain both supplied angles use the selected angle_unit; the calculator cannot infer that one input was written in degrees and the other in radians. Do not be alarmed when a textbook gives an angle different from quotient_argument by 360° or 2π. Those are equivalent polar descriptions, and argument_difference preserves the unnormalized subtraction for comparison. You can verify the rectangular result independently by checking that its magnitude matches quotient_modulus and that atan2(imaginary, real) matches the principal argument.
Integration use cases
Prepare deterministic test fixtures
Generate stable rounded outputs for software that accepts polar complex values and must be tested against rectangular results.
Technical questions
Can I use degrees or radians?
Yes. Set angle_unit to deg or rad and supply both input arguments in that same unit. Returned argument fields use the selected unit.
Why does the normalized angle differ from my subtraction?
Angles separated by a complete revolution describe the same complex number. argument_difference shows the raw subtraction, while quotient_argument shows an equivalent principal angle.
Can the denominator modulus be zero?
No. A zero modulus means the denominator is the complex number zero, and division by zero is undefined, so the request returns invalid_input.
What does an API request cost?
Each successful API request costs $0.002. The same deterministic core can also power the browser experience.